This project will be in the field of geometric graph theory, which is a link between graph theory and discrete geometry, and uses tools from both disciplines. Given a set S of line segments in the plane, we can construct a graph with a vertex for each line segment in S, and an edge between two vertices if and only if their corresponding segments intersect. Conversely, given a graph G, G is a segment intersection graph if it has such a representation with line segments. Research on segment intersection graphs has centered around Scheinerman's conjecture that all planar graphs are segment intersection graphs, stated in 1984 and proved by Chalopin and Goncalves in 2009. In this project, we'll investigate one or more variations of segment intersection graphs that are not as well studied. Possibilities include graphs represented by segments with k gaps, or with affine subspaces of n-dimensional space, or additional variations defined by the research group.

This project is focused in an area of graph theory called competitive graph coloring. The Graph Coloring Game has two players: Alice and Bob. The players alternate coloring the uncolored vertices of a finite graph G from a fixed finite set of colors. At each step of the game, the players must choose to color an uncolored vertex with a legal color. In the basic formation of the game, a color is legal for an uncolored vertex if the vertex has no neighbors already colored with that color. Alice wins the game if all vertices of the graph are colored; otherwise, Bob wins. The least r such that Alice has a winning strategy for this game on G is called the game chromatic number of G. If the definition of legal color is altered so that at each step in the game, each color class must have maximum degree at most d, for some fixed integer d ≥ 0, then the least r such that Alice has a winning strategy is called the d-relaxed game chromatic number of G. These parameters have been studied extensively for many classes of graphs. The classes for which the most is known are trees and forests. While upper bounds on the game chromatic number (and d-relaxed game chromatic number) are known for these classes, and the bounds are known to be achievable, it is not known what structural properties the graphs must have in order to achieve the bounds. This project will attempt to find these properties, thereby classifying these classes completely with respect to the Graph Coloring Game. We will then attempt to classify trees and forests relative to the d-relaxed game chromatic number for small values of d. Finally, we may consider other variations of the game in which the definition of legal color is altered in other ways.

The Game of Go: Statistical Approaches to Artificial Intelligence at Lewis and Clark College.

The Asian game of Go has simpler rules than Chess, but writing a Go-playing program that can compete with strong human players has proven exceedingly difficult. In fact, Go is considered one of the "grand challenges" of artificial intelligence. We will explore various statistical/machine learning approaches to the problem, including Monte Carlo methods and learning from recorded games. This summer's work will focus on decomposing the board into regions, so that the program can perform and combine several local searches (with the work possibly spread over several machines) rather than performing a single global search.

Desired skills:
- Programming experience, especially in Java
- Basic knowledge of statistics
- Experience playing Go
(Parallel programming experience would be a plus.)
Here's more on the project so far: Orego
If you are not familiar with Go, please see:
http://www.usgo.org/usa/waytogo/index.html

Pattern Recognition of Physical Activity Data at the University of Portland.

Accurately measuring behaviors critical to overall health, such as physical activity, has generally relied upon individualized self-report of recalled behavior. Recent adoption of technologies allowing for the accurate measurement of such a complex behavior has resulted in the need for advanced mathematical approaches to analyze the output. One specific device, the accelerometer, is a wearable device that monitors body acceleration in three planes, which is then processed and filtered to provide estimates of time spent engaging in different intensities of physical movement. The current project will begin with calibrating and validating data from accelerometers worn by subjects from a college-aged sample. This will include transforming the raw accelerometer data to activity signals that can be analyzed. We will then investigate an optimal method of pattern recognition to predict types of and total time spent in specific intensities of physically activity.

Prior background in probability, statistics, and programming will be helpful.